Quartiles for Screening, Not for Ranges

1. Why not industry averages

The 1968 regulations restricted the use of industry statistics under the resale price and cost plus methods to situations in which data on comparable uncontrolled transactions could not be found (Reg. §§ 1.482-2(e)(3)(vii) and 1.482-2(e)(4)(iv)). The Tax Court had added a judicial reason in Nissho Iwai (1985), holding industry averages to be, in most cases, of uncertain reliability for determining arm’s length prices. The 1991 report supplied the statistical reason behind the judicial one: industry averages were published without their standard errors, and they were computed from unfiltered data, disregarding asymmetry and outliers.

The report proposed an explicit admissibility criterion. A statistic is reliable when its standard error is small relative to the statistic itself; by convention, when the coefficient of variation does not exceed one-half:

\quad (1)\qquad \mathrm{CV}=\frac{\operatorname{SE}(\bar{x})}{\bar{x}}\le\tfrac{1}{2} \qquad\Longleftrightarrow\qquad t=\frac{\bar{x}}{\operatorname{SE}(\bar{x})}\ge 2

The industry publications then in use, Robert Morris Associates’ Annual Statement Studies and Dun and Bradstreet’s Industry Norms and Key Business Ratios, reported no standard errors. To their credit, both reported medians and quartiles rather than means, which the report acknowledged as recognition of the same problem. The operative conclusion was that statistics used for an adjustment must be admissible in court, and that industry ratios were not to serve as the initial method of adjustment or as a short-cut to the factual development of a case.

2. Data and ratios

The ratios were derived from the stratified samples of corporate income tax returns compiled each year by the Statistics of Income Division, tabulated by Principal Business Activity (PBA) code, a classification similar to but not identical with the SIC. The population was defined by ownership rather than by a vendor’s industry list: domestic corporations whose principal activity was wholesale trade and which had no foreign owner. This definition is the uncontrolled population in the sense the regulations require, and it is the definition a database of public filers cannot supply without a separate ownership test.

Five ratios were tabulated for eighteen PBA lines, ten in durable goods and eight in nondurable goods:

For each cell the report gives the sample size N, the lower quartile, the median and the upper quartile. N ranges from 77 (drugs and druggists’ sundries, advertising ratio, 1987) to 1,407 (machinery, equipment and supplies, Berry ratio, 1985). Reporting N beside each quartile is not a courtesy; the sample size is one of the two ingredients of the standard error that criterion (1) demands.

3. Resistance and the outlier rule

Preliminary analysis with stem-and-leaf displays, which unlike the histogram show every observation in the batch, revealed elongated tails with unusual ratios at both ends. Extreme values of this kind distort the sample mean and the standard deviation. Two remedies were available: report the median and quartiles, which are insensitive to extreme values, or isolate the outliers and delete them before the mean and standard deviation are computed (Chen, Gnanadesikan and Kettenring 1974). The report took the first course and published raw, unfiltered quartiles, leaving the examiner the means of applying the second.

The filter is Tukey’s. With Q_{1} and Q_{3} the lower and upper quartiles, the cutoff is

\quad (2)\qquad \tau=1.5\,(Q_{3}-Q_{1}), \qquad\text{fences at}\qquad Q_{3}+\tau\quad\text{and}\quad Q_{1}-\tau

A company whose net profit margin exceeds Q_{3}+\tau, or, in the situation of interest for deviant inbound distributors, falls below Q_{1}-\tau, was flagged as a candidate with potential arm’s length problems. The asymmetry is deliberate. The inbound distributor under examination is suspected of reporting too little profit, so the lower fence is the operative one; the upper fence serves the outbound case. The performance of this rule, and of its relatives, had been studied by Hoaglin, Iglewicz and Tukey (1986), and the stem-and-leaf and resistant-summary methods came from the Hoaglin, Mosteller and Tukey volume of 1983. David Hoaglin of Harvard was engaged by the IRS to evaluate these screening methods.

The point to retain is the role assigned to the interquartile range. It is the scale parameter of an outlier rule. The box between Q_{1} and Q_{3} is where screening stops; it is not where the arm’s length result is established. A company inside the box is not flagged; it is not thereby vindicated.

4. What the tables show

Two regularities in the tables have outlasted the report. First, the Berry ratio medians lie between 0.96 and 1.12 across all fifty-four cells (eighteen lines, three years), whereas the gross profit margin medians range from 10.5 to 28.2 percent. The gross margin is industry-specific; the Berry ratio is not. This follows from construction, since the Berry ratio divides gross profit by the operating expenses that gross profit must cover, and a median near unity states that the median wholesaler earned an operating profit of a few percent of sales. The net margin tables confirm it: medians between 0.5 and 2.0 percent in every line and year.

Second, the lower quartile of the net profit margin is negative in nine of the eighteen lines in 1987, in eleven lines in 1985 and in seven lines in 1986. The first quartile of raw SOI data contains loss-making companies. Any interval bounded below by Q_{1} of an unfiltered distribution admits, as comparables, enterprises that fail a going-concern test. The 1991 report did not propose such an interval; it proposed fences that lie 1.5\,\mathrm{IQR} outside it.

Appendix A reproduces the 1987 figures for all eighteen lines.

5. From screening to range

In July 1994, Reg. § 1.482-1(e)(2)(iii)(B) provided that the interquartile range ordinarily establishes the arm’s length range when the comparables are of imperfect quality, while permitting a more reliable statistical method where one can be shown. The same two order statistics, Q_{1} and Q_{3}, were thereby moved from the perimeter of an outlier rule to the boundaries of a presumptive arm’s length interval. Three distinctions separate the 1991 use from the 1994 one.

Dispersion versus estimation. The IQR measures cross-company variation and does not shrink as N grows; under normality,

\quad (3)\qquad \mathrm{IQR}=2\times 0.6745\,\sigma\approx 1.349\,\sigma

whatever the sample size, whereas a confidence interval for a parameter shrinks with \sqrt{N}. The 1991 report used the IQR for what it measures, the spread of a batch; the 1994 regulation uses it as if it bounded an estimate.

Ratios carry the intercept. If operating profit obeys Y_i=\alpha+\beta X_i in sales X_i, every profit margin equals

\quad (4)\qquad m_i=\frac{Y_i}{X_i}=\beta+\frac{\alpha}{X_i}, \qquad\text{so that}\qquad \mathrm{IQR}(m)=|\alpha|\left[\frac{1}{Q_{25}(X)}-\frac{1}{Q_{75}(X)}\right]

The quartiles of m inherit the term \alpha/X_i. The IQR of ratios does not test whether \alpha=0, and when \alpha\ne 0 the quartiles of m are not quartiles of \beta.

Admissibility. Criterion (1) requires a standard error. A quartile reported without one, like the industry average it replaced, cannot meet the test the report set for statistics used in adjustment. Regression on company-year data estimates the slope \beta, tests the intercept \alpha, and reports both with their standard errors; it is the more reliable method the regulation itself contemplates.

6. Two further objections: width and the target

The first objection is quantitative, and the 1991 tables measure it. For machinery wholesalers in 1987 the pretax-margin box runs from -0.13 to 3.59 percent of sales around a median of 1.33; for sporting goods, from -1.35 to 4.45 around 1.60. An interval that admits a loss at one end and three times the median at the other does not constrain a controlled price; it ratifies whatever the taxpayer reports. The width is not an accident. A box constructed so that outlier fences fall 1.5\,\mathrm{IQR} beyond its edges must be wide. Width was a virtue for screening and became a defect the moment the box was promoted to an adjustment rule. The one-standard-error interval for the regression slope, \hat{\beta}\pm\operatorname{SE}(\hat{\beta}), narrows with \sqrt{N}, which is what an adjustment instrument should do as evidence accumulates.

The second objection is structural. Management does not target an interval. It targets a markup on cost or a rate of return on sales, capital or assets, the point documented in the Brookings surveys of administered pricing and formalized in the Kaleckian price equation. A dispersion measure across heterogeneous comparables describes how far other enterprises miss their own targets; it says nothing about the target the tested party should have attained. The regulations concede this at the moment of adjustment: when the tested party’s result falls outside the box, the adjustment is to the median, a point. If a point suffices for the adjustment, the interval was never the operative object. It was a tolerance band, and a tolerance band wants a standard error, not two order statistics.

None of this is new to readers of this blog. What the 1991 report adds is provenance. Quartiles entered transfer pricing practice at the IRS as a resistant screening device, chosen because the data were asymmetric and contaminated with outliers, accompanied by a reliability criterion and a warning against using industry statistics as a method of adjustment. Their later elevation into a presumptive range was a change of function, not a development of the original method. The 1991 report put the quartiles to the use their construction warrants.

References

Chen, H., R. Gnanadesikan and J. R. Kettenring (1974), “Statistical Methods for Grouping Corporations,” Sankhyā, Series B, 36(1), 1–28. Source of the isolate-and-delete alternative to resistant summaries.

Emerson, J. D. and D. C. Hoaglin (1983), “Stem-and-Leaf Displays,” in Hoaglin, Mosteller and Tukey (1983), ch. 1. The display that showed every observation in the SOI batches.

Hoaglin, D. C., B. Iglewicz and J. W. Tukey (1986), “Performance of Some Resistant Rules for Outlier Labeling,” Journal of the American Statistical Association, 81(396), 991–999. Evaluates the 1.5\,\mathrm{IQR} fence and its relatives.

Hoaglin, D. C., F. Mosteller and J. W. Tukey, eds. (1983), Understanding Robust and Exploratory Data Analysis, New York: John Wiley. The methodological source of the 1991 report.

Internal Revenue Service, Statistics of Income Division, Corporation Income Tax Returns, 1985, 1986, 1987, section 3 (Description of the Sample and Limitations of the Data).

Lanzillotti, R. F. (1958), “Pricing Objectives in Large Companies,” American Economic Review, 48(5), 921–940. The Brookings survey evidence that management targets a rate of return.

Nissho Iwai American Corp. v. Commissioner, 50 T.C.M. (CCH) 1483 (1985). Industry averages held to be of uncertain reliability for arm’s length pricing.

Silva, E. A. (1991), Corporate Statistics of Income: Industry Ratios for Wholesale Trade in the U.S., 1985–1987, Internal Revenue Service, June 26, 1991. The report discussed in this note.

Tukey, J. W. (1977), Exploratory Data Analysis, Reading, Mass.: Addison-Wesley. Origin of the box plot, the fences and the 1.5\,\mathrm{IQR} convention.

Treas. Reg. § 1.482-1(e)(2)(iii)(B) (1994); Treas. Reg. §§ 1.482-2(e)(3)(vii) and 1.482-2(e)(4)(iv) (1968).